Culture secretary Jeremy Hunt is under pressure to explain his links to News Corporation after it was alleged that he secretly backed the company's bid to take control of BSkyB.

Emails handed to the Leveson inquiry into press ethics suggest Hunt's office was communicating with News Corporation while he was charged with impartially adjudicating on the deal.

That power had initially been in the hands of Lib Dem business secretary Vince Cable, but he was stripped of the role in December 2010 after he was secretly filmed saying he was against the deal and that he had “declared war on Murdoch”.

David Cameron then handed the quasi-judicial responsibility for examining the deal to Hunt.

The Leveson inquiry heard on Tuesday that Frederic Michel, News Corporation's head of public affairs, told his bosses in Wapping that Hunt had a favourable view of the company's desire to take over the broadcaster.

In one message Michel detailed what the Hunt would say to parliament the next day, noting that it was "absolutely illegal" for him to obtain the information.

Another email, dating from January last year, reported Hunt's belief that it would be "game over" for opponents of the BSkyB takeover once plans to spin off Sky News into a separately listed company were publicly announced.

"He said we would get there at the end, and he shared our objectives," Michel noted.

Emails handed to the inquiry also suggested frequent contact between one of Hunt's special advisers and News Corporation.

Ed Miliband said Hunt had been shown to be "acting in the interests of the Murdochs, not the British people" and should resign.

"He himself said that his duty was to be transparent, impartial and fair in the BSkyB takeover. But now we know that he was providing advice, guidance and privileged access to News Corporation. He was acting as a backchannel for the Murdochs," he said.

"He cannot stay in his post. And if he refuses to resign, the prime minister must show some leadership and fire him."

And Harriet Harman, the party's deputy leader, said Hunt had fallen "woefully short of the standards expected by this office and the public interest".

Labour MP Tom Watson, who has long pursued allegations of phone hacking by News International publications, said it had been the "most incredible hour of evidence before Leveson so far".

"It paints a picture of a web of connections between ministers and their advisers, and News International's executives and lobbyists," he told The Huffington Post UK.

"The revelations of internal emails showing the role of Jeremy Hunt are truly shocking. I have no doubt parliament will want to look at his behaviour in detail."

"It's no surprise to me that the odds on him being the next minister to leave to leave the government have been slashed.

"It makes a mockery of all those MPs that questioned the legitimacy of the PM appointing Jeremy Hunt to make this quasi-judicial decision."

But James Murdoch, in his evidence before the hearing, denied that Hunt was "cheerleader" for News Corporation inside the government.

And he rejected the suggestion that the decision by The Sun to switch its support to the Tories before the 2010 general election meant News Corporation was more likely to secure the deal.

"That is absolutely not the case and the question of support of an individual newspaper for politicians one way or another is not something that I would ever link to a commercial transaction like this," he said.

The revelation that Hunt's office was communicating with News Corporation in relation to the takeover has led to speculation that the culture secretary may have to resign, with bookmakers including Ladbrooks suspending betting on his future.

However according to the BBC Hunt is "not even considering resignation" and will give his own side of the story to the Leveson inquiry in the coming days.

The inquiry heard there were a number of meetings between News Corp representatives, including Murdoch, and various government figures, including Cable and Culture Secretary Jeremy Hunt, during the company's offer for BSkyB.

Murdoch and Brooks met Tory MP Oliver Letwin in July 2009, and he met Hunt in October 2009 and February 2010 when they discussed issues relating to the media including piracy, broadband, Ofcom, and media regulation.

But he told the hearing the company had not gained any advantage from its communications with politicians, although there may have been a perception by politicians that support from a newspaper like the Sun would help them.

The inquiry heard that consultation was being carried out on the bid which was due to finish on July 8 2011 - but on July 4 that year the Guardian published its story about Milly Dowler's phone being hacked. It eventually prompted News Corp to withdraw the bid.